Semiperfect numbers

A semiperfect/pseudoperfect number is an integer equal to the sum of a part of or all of its divisors (excluding itself). Numbers which are equal to the sum of all of their divisors are perfect.

Divisors of 6 : 1,2,3
      6 = 1+2+3 -> semiperfect (perfect)
Divisors of 28 : 1,2,4,7,14
      28 = 14+7+4+2+1 -> semiperfect (perfect)
Divisors of 40 : 1,2,4,5,8,10,20
      40 = 1+4+5+10+20 or 2+8+10+20 -> semiperfect


A primitive semiperfect number is a semiperfect number with no semiperfect divisors (except itself :))

Divisors of 6 : 1,2,3
      6  = 1+2+3 -> primitive
Divisors of 12 : 1,2,3,4,6
      12 = 2+4+6 -> semiperfect

As references, please use the OEIS series A006036 for primitive semiperfect numbers, and A005835 for semiperfects.


Write a program or a function in any language. It will take as input a number n as a function parameter or from STDIN/your language's closest alternative, and will output all the primitive semi-perfect numbers from 1 to n (inclusive).

The output must be formated as 6[separator]20[separator]28[separator]88... where [separator] is either as newline, a space or a comma. There must not be a starting [separator] nor a ending one.

Edit : you can leave a trailing newline


input :


output :

input :


output :


input :


output :

6 20 28 88


This is code-golf, so the shortest code in bytes win.

Don't try to fool us with loopholes please :).

I'd be glad you could leave an explanation of your golfed code once you think you're done golfing it!

As this challenge already have some nice answers, and is slowly getting quiet, I'll put an end to it. The winner of this code-golf will be decided on Monday 29th, 00:00 GMT. Well done to all of you that have answered, and good luck to those who will try to beat them :)


6 Answers 6


Pyth, 28 27 bytes


1 byte thanks to @Jakube


                                Y = []
                                Q = eval(input())
VQ                              for N in range(Q):
    KhN                         K = N+1
           f    SN              filter T over range(1, N)
            !%KT                the logical not of K%T.
                                This is the list of divisors of K.
          J                     Store the list in J.
         y                      Create all of its subsets.
       sM                       Map each subset to its sum.
  I}K                           If K is in that list: (If K is semiperfect)
                  I!@JY         If the intersection of J (the divisors)
                                and Y (the list of primitive semiperfect numbers)
                                is empty:
                        aYK     Append K to Y
                       e        And print its last element, K.
  • \$\begingroup\$ @AlexA. Thanks! It is necessary to append K to Y to build Y, which is needed elsewhere. However, I could do the printing separately, such as with aYKK instead of eaYK. It's 4 bytes either way, however. \$\endgroup\$
    – isaacg
    Jun 19, 2015 at 15:03

Julia, 161 149 bytes

n->(S(m)=!isempty(filter(i->i==unique(i)&&length(i)>1&&all(j->m%j<1,i),partitions(m)));for i=2:n S(i)&&!any(S,filter(k->i%k<1,1:i-1))&&println(i)end)

This creates an unnamed function that accepts an integer as input and prints the numbers to STDOUT separated by a newline. To call it, give it a name, e.g. f=n->....

Ungolfed + explanation:

# Define a function that determines whether the input is semiperfect
# (In the submission, this is defined as a named inline function within the
# primary function. I've separated it here for clarity.)

function S(m)
    # Get all integer arrays which sum to m
    p = partitions(m)

    # Filter the partitions to subsets of the divisors of m
    d = filter(i -> i == unique(i) && length(i) > 1 && all(j -> m % j == 0, i), p)

    # If d is nonempty, the input is semiperfect

# The main function

function f(n)
    # Loop through all integers from 2 to n
    for i = 2:n
        # Determine whether i is semiperfect
        if S(i)
            # If no divisors of i are semiperfect, print i
            !any(S, filter(k -> i % k == 0, 1:i-1) && println(i)


julia> f(5)

julia> f(40)

JavaScript (ES6) 172

Run the snippet below to test


// Less golfed

   h=[]; // hashtable with numbers found so far

   for (n=1; n <= v; n++)
      r=[1],l=1; // r is the list of divisors, l is the length of this list
      t=1; // used as a flag, will become 0 if a divisor is in h
      for(i=2; i<n; i++)
         if (n%i == 0)
            if (h[i])
               t = 0; // found a divisor in h, n is not primitive
               l = r.push(i); // add divisor to r and adjust l
      if (t != 0) // this 'if' is merged with the for below in golfed code
         // try all the sums, use a bit mask to find combinations
         for(i = 1; t != 0 && i < 1<<l; i++)
            t = n; // start with n and subtract, if ok result will be 0 
            m = 0.5; // start with mask 1/2 (nice that in Javascript we can mix int and floats)
            r.forEach( v=> i & (m+=m) ? t -= v : 0);
         if (t == 0 && n > 1) h[n] = 1; // add n to the hashmap (the value can be anything)
   // the hashmap keys list is the result
   return '' + Object.keys(h) // convert to string, adding commas

(test=()=> O.textContent=f(+I.value))();
<input id=I type=number oninput="test()" value=999><pre id=O></pre>


CJam, 54 bytes

This solution feels a bit awkward, but since there have been few answers, and none in CJam, I thought I'd post it anyway:


A good part of the increment over the posted Pyth solution comes from the fact that, as far as I could find, CJam does not have an operator to enumerate all subsets of a set. So it took some work to complete that with available operators. Of course, if there actually is a simple operator I missed, I'll look kind of silly. :)


L     Start stack with empty list that will become list of solutions.
li    Get input N and convert to int.
),2>  Build list of candidate solutions [2 .. N].
{     Start for loop over all candidate solutions.
_     Copy list of previous solutions, needed later to check for candidate being primitive.
N,1>  Build list of possible divisors [1 .. N-1].
{N\%!},  Filter list to only contain actual divisors of N.
_     Check if one of divisors is a previous solution. Start by copying divisor list.
@     Pull copy of list with previous solutions to top of stack
&!    Intersect the two lists, and check the result for empty. Will be used later.
\     Swap top two elements, getting divisor list back to top.
_,    Get length of divisor list.
2,    Put [0 1] on top of stack.
m*    Cartesian power. Creates all 0/1 sequences with same length as divisor list.
\     Swap with divisor list.
f{.*:+}  Calculate element by element product of all 0/1 sequences with divisors,
         and sum up the values (i.e. dot products of 0/1 sequences with divisors).
         The result is an array with all possible divisor sums.
N#)  Find N in list of divisor sums, and covert to truth value.
e&   Logical and with earlier result from primitive test.
{N+}&  Add N to list of solutions if result is true.
}fN  Phew! We finally made it to the end of the for loop, and have a list of solutions.
S*   Join the list of solutions with spaces in between.

Try it online


PHP, 263 Bytes

function m($a,$n){for($t=1,$b=2**count($a);--$b*$t;$t*=$r!=$n,$r=0)foreach($a as$k=>$v)$r+=($b>>$k&1)*$v;return$t;}for($o=[];$i++<$argn;m($d,$i)?:$o=array_merge($o,range($r[]=$i,3*$argn,$i)))for($d=[],$n=$i;--$n*!in_array($i,$o);)$i%$n?:$d[]=$n;echo join(",",$r);

Try it online!


function m($a,$n){ 
  for($t=1,$b=2**count($a);--$b*$t;$t*=$r!=$n,$r=0) #loop through bitmasks
    foreach($a as$k=>$v)$r+=($b>>$k&1)*$v; # loop through divisor array
  return$t;} # returns false for semiperfect numbers 
  :$o=array_merge($o,range($r[]=$i,3*$argn,$i))) # Make the result array and the array of multiples of the result array 
  for($d=[],$n=$i;--$n*!in_array($i,$o);) # check if integer is not in multiples array
    $i%$n?:$d[]=$n; # make divisor array
echo join(",",$r); #Output

Jelly, 22 bytes


Try it online!


ÆDṖŒPS€i - helper function to check if input is a semiperfect number
ÆD       - list of divisors of input
  Ṗ      - except for the last one (the input)
   ŒP    - power set = every possible subset of divisors
     S€  - sum of each subset
       i - return truthy iff input is one of these

ÆDÇ€TL’ - helper function to check if input is a primitive semiperfect number
ÆD       - list of divisors of input
  ǀ     - replace each with if they are a semiperfect number, based on 
           the above helper function. If input is a primitive semiperfect 
           number, we get something like [0,0,0,0,0,94]. 
    T    - get all truthy values.
     L’  - return falsy iff there is only one truthy value

RÇÐḟY    - main link
R        - Range[input]
 ÇÐḟ     - Filter out those elements which are not primitive semiperfect
           numbers, based on the helper function
    Y    - join by newlines.

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